Table Of ContentUniversitext
Editorial Board
(North America):
S. Axler
F.W. Gehring
K.A. Ribet
Springer Science+B usiness Media, LLC
U niversitext
Editors (North America): S. Axler, F.W. Gehring, and K.A. Ribet
AksoylKhamsi: Nonstandard Methods in Fixed Point Theory
Andersson: Topics in Complex Analysis
Aupetit: A Primer on Spectral Theory
BachmanINariciIBeckenstein: Fourier and Wavelet Analysis
BalakrishnanlRanganathan: A Textbook of Graph Theory
Balser: Fonnal Power Series and Linear Systems of Merom orphic Ordinary
Differential Equations
Bapat: Linear Algebra and Linear Models (2nd ed.)
Berberian: Fundamentals of Real Analysis
BoossIBleecker: Topology and Analysis
Borkar: Probability Theory: An Advanced Course
BiittcherlSilbermann: Introduction to Large Truncated Toeplitz Matrices
CarlesonlGamelin: Complex Dynamics
Cecil: Lie Sphere Geometry: With Applications to Submanifolds
Chae: Lebesgue Integration (2nd ed.)
Charlap: Bieberbach Groups and Flat Manifolds
Chern: Complex Manifolds Without Potential Theory
Cohn: A Classical Invitation to Algebraic Numbers and Class Fields
Curtis: Abstract Linear Algebra
Curtis: Matrix Groups
DiBenedetto: Degenerate Parabolic Equations
Dimca: Singularities and Topology of Hyper surfaces
Edwards: A Formal Background to Mathematics I alb
Edwards: A Formal Background to Mathematics II alb
Foulds: Graph Theory Applications
Friedman: Algebraic Surfaces and Holomorphic Vector Bundles
Fuhrmann: A Polynomial Approach to Linear Algebra
Gardiner: A First Course in Group Theory
GardingITambour: Algebra for Computer Science
Goldblatt: Orthogonality and Spacetime Geometry
GustafsonlRao: Numerical Range: The Field of Values of Linear Operators
and Matrices
Hahn: Quadratic Algebras, Clifford Algebras, and Arithmetic Witt Groups
Heinonen: Lectures on Analysis on Metric Spaces
Holmgren: A First Course in Discrete Dynamical Systems
Howe!fan: Non-Abelian Harmonic Analysis: Applications of SL(2, R)
Howes: Modem Analysis and Topology
Hsieh/Sibuya: Basic Theory of Ordinary Differential Equations
HumiIMiller: Second Course in Ordinary Differential Equations
HurwitzlKritikos: Lectures on Number Theory
Jennings: Modem Geometry with Applications
JonesIMorrisIPearson: Abstract Algebra and Famous Impossibilities
KannanIKrueger: Advanced Analysis
KeIlylMatthews: The Non-Euclidean Hyperbolic Plane
Kostrikin: Introduction to Algebra
LueckingIRubel: Complex Analysis: A Functional Analysis Approach
MacLaneIMoerdijk: Sheaves in Geometry and Logic
Marcus: Number Fields
(continued after index)
Juha Heinonen
Lectures on
Analysis on
Metric Spaces
i
Springer
Juha Heinonen
Mathematics Department
East HaU
University of Michigan
Ann Arbor, MI 48109-1109
USA
Editorial Board
(North America):
S. Axler F.W. Gehring
Mathematics Department Mathematics Department
San Francisco State University East Hali
San Francisco, CA 94132 University of Michigan
USA Ann Arbor, MI 48109-1109
USA
K.A. Ribet
Mathematics Department
University of California at Berkeley
Berkeley, CA 94720-3840
USA
Mathematics Subject Classification (2000): 28Axx, 43A85, 46Exx
Library of Congress Cataloging-in-Publication Data
Heinonen, Juha.
Lectures on analysis on metric spaces / Juha Heinonen.
p. cm - (Universitext)
Inc1udes bibliographical references and index.
ISBN 978-1-4612-6525-2 ISBN 978-1-4613-0131-8 (eBook)
DOI 10.1007/978-1-4613-0131-8
1. Metric spaces. 2. Mathematical analysis. 1. Title. II. Series.
QA611.28.H44 2001
514'.32--dc21 00-056266
Printed on acid-free paper.
© 2001 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc in 2001
Softcover reprint of the hardcover lst edition 2001
All rights reserved. This work may not be translated or copied in whole or in part without
the written permission ofthe publisher (Springer Science+Business Media, LLC), except
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Production managed by Michael Koy; manufacturing supervised by Jerome Basma.
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9 8 7 6 5 4 3 2 1
ISBN 978-1-4612-6525-2
To my teacher Professor Olli Martio
Preface
Analysis in spaces with no a priori smooth structure has progressed to include con
cepts from first-order calculus. In particular, there have been important advances
in understanding the infinitesimal versus global behavior of Lipschitz functions
and quasiconformal mappings in rather general settings; abstract Sobolev space
theories have been instrumental in this development. The purpose of this book is
to communicate some of the recent work in the area while preparing the reader to
study more substantial related articles.
The book is a written version of lectures from a graduate course I taught at
the University of Michigan in Fall 1996. I have added remarks and references to
works that have appeared thereafter. The material can roughly be divided into three
different types: classical, standard but sometimes with a new twist, and recent. For
instance, the treatment of covering theorems and their applications is classical,
with little novelty in the presentation. On the other hand, the ensuing discussion
on Sobolev spaces is more modern, emphasizing principles that are valid in larger
contexts. Finally, most ofthe material on quasi symmetric maps is relatively recent
and appears for the first time in book format; one can even find a few previously
unpublished results. The bibliography is extensive but by no means exhaustive.
The reader is assumed to be familiar with basic analysis such as Lebesgue's
theory of integration and the elementary Banach space theory. Some knowledge
of complex analysis is helpful but not necessary in the chapters that deal with
quasi symmetric maps. Occasionally, some elementary facts from other fields are
quoted, mostly in examples and remarks.
The students in the course, many colleagues, and friends made a number of
useful suggestions on the earlier versions of the manuscript and found many errors.
I thank them all. Special thanks go to four people: Bruce Hanson carefully read
viii Preface
the entire manuscript and detected more errors than anyone else; years later, use of
Kari Hag's handwritten class notes preserved the spirit of the class; Fred Gehring
insisted that I prepare these notes for publication; and Jeremy Tyson provided
crucial help in preparing the final version of the text. Finally, I am grateful for
having an inspiring audience for my lectures at Michigan.
Juha Heinonen
Ann Arbor, Michigan
July 1999
Contents
Preface vii
1. Covering Theorems . 1
2. Maximal Functions . 10
3. Sobolev Spaces . .. 14
4. Poincare Inequality . 27
5. Sobolev Spaces on Metric Spaces 34
6. Lipschitz Functions . . . . . . . . 43
7. Modulus of a Curve Family, Capacity, and
Upper Gradients 49
8. Loewner Spaces . 59
9. Loewner Spaces and Poincare Inequalities 68
10. Quasisymmetric Maps: Basic Theory I 78
11. Quasisymmetric Maps: Basic Theory II 88
12. Quasisymmetric Embeddings of Metric Spaces
in Euclidean Space . . . . . . . . 98
13. Existence of Doubling Measures 103
x Contents
14. Doubling Measures and Quasisymmetric Maps .... 109
IS. Conformal Gauges . . . . . . . . . . . . . . . . . . . . 119
References 127
Index 137
1
Covering Theorems
All covering theorems are based on the same idea: from an arbitrary cover of a
set in a metric space, one tries to select a subcover that is, in a certain sense, as
disjointed as possible. In order to have such a result, one needs to assume that the
covering sets are somehow nice, usually balls. In applications, the metric space
normally comes with a measure /-L, so that if F = {B} is a covering of a set A by
balls, then always
/-L(A)::: L/-L(B)
F
(with proper interpretation of the sum if the collection F is not countable). What
we often would like to have, for instance, is an inequality in the other direction,
/-L(A) 2: C L /-L(B),
F'
for some subcollection F' c F that still covers A and for some positive constant C
that is independent of A and the covering F. There are many versions of this theme.
In the following, we describe three covering theorems, which we call the basic
covering theorem, the Vitali covering theorem, and the Besicovitch-Federer cov
ering theorem. (The terms may not be standard.) Each of these three theorems has
a slightly different purpose, as well as different domains of validity, and I have
chosen them as representatives of the many existing covering theorems. We will
not strive for the greatest generality here.
1.1 Convention. Let X be a metric space. When we speak of a ball B in X, it is
understood that it comes with a fixed center and radius, although these in general
J. Heinonen, Lectures on Analysis on Metric Spaces
© Springer Science+Business Media New York 2001
